We consider L-infinity quasi-isomorphisms for Hochschild cochains whose structure maps admit ``graphical expansion´´. We introduce the notion of stable formality quasi-isomorphism which formalizes such an L-infinity quasi-isomorphism. We define a homotopy equivalence on the set of stable formality quasi-isomorphisms and prove that the set of homotopy classes of stable formality quasi-isomorphisms form a torsor over the group corresponding to the zeroth cohomology of the full graph complex. This result may be interpreted as a full description of homotopy classes of formality quasi-isomorphisms for Hochschild cochains in the ``stable setting´´.